I drove this bridge fairly regularly when I lived in the bay. I've personally never seen the actual bridge itself slowed down, it's mostly the north side entrance failing to understand how to zipper merge and backing up to the 101. The south side will also back up but that's also before the bridge and more importantly its due to the 10 lane toll booth right before the entrance.
Opening a lane on the south side will not fix this issue like it did on the north side, (Due to the toll road and the major merge a mile up the road). you're essentially shutting down access to east bay for thousands of bicyclists for a fix that frankly isn't needed and wont work.
https://www.zermatt.ch/en/sustainability/Elektros-Autofrei-Anreise/Zermatt-is-car-free
while you could argue the specialized electric buses and delivery vans are "Cars" they cannot drive on normal roads and do not produce pollution, in terms of emissions or noise, like regular cars. It's more than possible to build a car free city, its been done before.